The shortest distance between a point and a line is the length of the line segment that is perpendicular to the given line. We will need to find the perpendicular line and then we can find the intersection point. Finally, we can calculate the distance from the given point to the point of intersection.
Finding the Perpendicular Line
Perpendicular lines have negative reciprocal slopes. This means the product of their slopes is equal to -1.
m_1 * m_2=- 1
Examining the slope-intercept form of the line representing the nature trail, we know that it has a slope of 13. By substituting this value into the formula, we can find the slope of the perpendicular line.
The slope of the perpendicular line is - 3. To find the equation of the line, we substitute the known point in the slope-intercept form, setting the slope to - 3.
The equation of the perpendicular line is
y=- 3x-14
Finding the Point of Intersection
To find our distance, we also need to know where the given line and the perpendicular line intersect. By setting up a system of equations, we can find the point of intersection.
y=- 3x-14 y= 13x-4
Since both equations have y isolated, it's convenient to use the Substitution Method.
The distance from the point (- 6,4) to the line y=x+4 is sqrt(90).
Since each unit in the coordinate plane represents 10 feet, we find the total distance by multiplying the unit distance by the distance between the points.
10* sqrt(90)≈ 94.9 feet
